Trigonometrical protractor.



D. AGAPOFF. TRIGONOMETRICAL PROTRACTOR. APPLICATION FILED APR. 25. 1914.

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D. AGAPOFF.

TRIGONOMETRICAL PROTRACTOR.

APPLICATION FILED APR.25, I914.

Patented Oct. 17, 1916.

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- DMITRI AGAPOFF, 0F ORENBURG, RUSSIA.

TRIGONOMETRICAL rnornao'ro'n.

Specification of Letters Patent.

Patented oer. it, 1916.

Application filed. April 25, 1914. Serial No. 834,471.

I '0 all whom it may concern Be it known that I, DMITRI AGAPOFF, a citizen of the Russian Empire, residing at Orenburg, in the Empire of Russia, have invented certain new and useful Improvements in Trigonometrical Protractors, of

which the following is a specification.

By means of the trigonometrical protractors already known the trigonometrical values of angles between 0 and only can be directly read. I

The subject-matter of my invention is a trigonometrical protractor which provides for the direct reading of the trigonometrical values of all angles between 0 and 180 and substantially consists of isosceles congruent right-angled triangles, the side of one triangle coinciding with part of the hypotenuse of the other the hypotenuses of the triangles meeting in a point and the side of each triangle located opposite said point being uniformly divided so that it is possible to read at any point on the scales the tangent of each angle which is inclosed by the other side and the straight line joining the point on the scale and the said point.

.One illustrative embodiment of my invention and two modifications thereof are represented by way 'of example in the accompanying drawings, wherein:

Figure 1 is a plan view showing one form of my trigonometrical protract-or, Fig. 2 is a like view showing a modified form, and Fig. 3 is a like view showing another form comprising an alidade and verniers, Fig. 4c is a geometricalfigure for explaining the principles on which the invention is based, Fig. 5 is a fragmentary detail sectional view showing the method of mounting the vernier, and taken on the line 1--1 of Fig. 3.

Referring first to Fig. f it is known that in an isoscele, right-angled triangle ABC each acute angle is 45 and its tangent is.

equal to 1. If the side located opposite the angle CBA is divided into three equal parts and if a straight line is drawn from each point of division K and M to the vertex B of the angle, three angles ABK, ABM and ABC are obtained, the ratios of the lengths AK, AM and AC to the length AB being the tangent of the corresponding angle. It is seen that as many tangents of the angles between 0 and 45 are obtained as there are equal lengths into which the side AG is divided. If a length BDzAC is measured along the hypotenuse BC and a perpendicular DEzBDzAG is erected at the point D and if this straight line DE is likewise divided into three equal parts and the points of division K and M are connected with the point B, three equal angles DBK ABK, DBM ABM and DBE ABC are obtained. Then any angle of the angle DBE at the vertex B can be added to the angle CBA and the corresponding tangent of an angle more than 45 namely an angle oz+l5 can be found by the known formula tan. a+tan. 5

tan. +5) fl'tf fi fa tan. 5

thus

- tan. CBK +tan. CBA OBK1 LABQ 1 tan. one; tan. CBA

In such a manner it is possible to determine directly the tangent of a given angle between 0 and 90 inclusive. of angles exceeding 90 can be at once found as the tangent of any given obtuse angle is equal to the tangent of the acute angle which is a supplement of the obtuse angle to 180 with that difference only that the value is a negative one.

The construction of my improved trigonometrical protractor is based on the above described considerations. From the above it will be understood that the following conditions must be fulfilled by this trigonometrical protractor: The length of the outside sides of the protractor must be equal, 91. e.

AB AC DE, the angles CBAzACB:

DBE BED must equal one another and be 45. The angles BAC BDE must be equal to one another and be 90, and the straight lines AC and DE must be uniformly divided into equal parts.

To this conditions corresponds the protractor shown in Fig. 1 and consists of two pairs of connected isosceles right-angled congruent triangles OGH, OJ L and OLP, OQR which together form one entirety. The outermost sides GH, JL, LP, RQ, which must be equal to one another, are divided into a number of equal parts in such manner that the instrument will read with an accuracy of 0.01. The use of this instrument enables the tangent of any angle between 0 and 45 inclusive to be determined directly. The tangent of angles exceeding 45 can be at once found with the aid of the The tangent above mentioned formula for the total of two angles, especially as the one angle is 45 and its tangent l. v

The protractor represented in Fig. 2 differs from that just described only in the scales of the sides J L and PL being marked with the corresponding values of the tangents, so that the tangent of any angle from 0 to 90 inclusive can be read with an accuracy of 0.02. 1

Fig. 3 shows a protractor having an alidade S mounted to rotate around the point 0, and vernier T by which the tangent of any angle can be read with an accuracy of 0.001.

The verniers T, as indicated in Fig. 5, are

" slidably mounted on the graduated edges of their respective portions oifthe protractor plate by means of a dove-tailed finger t which takes into a eomplementarilv shaped groove t on the under side of the graduated edge. This methodof attachment effectively prevents any lateral disarrangement of the Vernier and'at t-helsaine time permits of the easy adjustment along the graduated edge.

4 In the example shown in Fig; 3 the alidade I S is shown located between the graduations 0.36 and 0.37, and its edge Z is supposed to" be resting against one sideof the angle to be measured, the other sideof such angle coinciding with the line RO. As shown the fifth division of the Vernier coincideswith one of the graduations on the line PL, thus indicating a fraction of five-tenths of one of the graduations or a total reading of .365. Therefore the tangent of the angle ROZ is greater than and the formula would read,

For the purpose of measuring angles Gopiespfthis patent may be obtained for five cents each, by addressing the 3. There are two such second sets of gradnations, one for each of the sides PL and JL. These second sets of graduations are made to read directly in tangents, respectively, of the angles included within ROL or GOL. It is therefore obvious, that with my improved form of protractor it is possible to make direct readings of all angles from 0 to 180.

I claim:

1. A trigonometrical protractor divided into isosceles, congruent, rightsangled triangles the side of one triangle coinciding with partot the hypotenuse of the other the hypotenuses of the triangles meeting in a point and each side of each triangle located opposite said point being uniformly divided.

2. A trigonometrical protractor divided into two pairs of oppositely arranged triangles integral one with another, each pair of triangles consisting of two congruent, isosceles,"right-angled triangles having the side of one triangle coinciding with part of the hypotenuse of the other, the hypotenuses of all the triangles meeting in a point, and

the side of each triangle located opposite said point being divided to read in tangents.

3. A trigonometrical protractor divided into two pairs of oppositely arranged triangles integral one with another, each pair of triangles consisting of two congruent, isosceles, right-angled triangles having the sideof one triangle coinciding with part of the hypotenuse of the other, the hypotenuses of all the triangles meeting in a point, and the side of each triangle located opposite said point being divided to read in tangents, an alidade rotatable about'said point over the divided sides of the triangles, and verniers each movable along one of the divided sides of the triangles.

In testimony whereof I have afiixed my signature in presence of two witnesses.

DMITRI AGAPOFF.

Washington, I). C. 

